If I change the SA password for a Microsoft SQL Server, can I see in a log or a trace if some application somewhere is trying to log in with the old password?
See this article/tutorial which explains how to enable login auditing in SQL Server 2000 and 2005:
The native audit level configuration is a simple configuration that can help to determine if the sa login is used in the SQL Server instance. What this configuration does is records all of the successful (or failed or both) logins into a particular SQL Server instance. You will not know the host name, application, etc. However, this configuration will enable you to determine if the sa login is being used in the first place. Once you have this information, then you can dig deeper with SQL Server Profiler outlined below.
I would tend to agree with the change it and see who complains, BUT only after some due diligence. You have to spend the time to profile and look for someone logging in with SA. If you can't find anything logging in over a week, then you ought to be safe. If it's used less often, it's even more of a reason to make the change.
No one ever wants to change this password, and it ends up creating issues. The SA password gets out of control and when it is disclosed to the wrong person, it creates havoc. Often with data loss, someone losing data with a lack of backups, new database, DELETE with no WHERE, etc.
Disclose you are making the change, run Profiler, and then do it. Your customers will appreciate it.